Mini Blackjack and Card Counting

Prerequisites: Chapters 1, 3, 4, 5.

Do you play in the Casino? Would you? Why not? If your answer is that Casino games are designed in such a way that the Casino has higher odds, you are only partially right. Black Jack is one of the few casino games where players, if playing in a sophisticated way, can actually win money in the long run. In the next chapter we will hear some stories about these attempts of analyzing the game. For those of you eager to go out yourself and break the Casino, you will be disappointed to hear that we are not discussing Black Jack in detail in this book for two reasons: First, it is a little too complicated, having some extra rules, and secondly and more important, it has a very large extensive form. A mini version, named "MINI BLACKJACK", however has all ingredients of the real one, but is simple enough to be analyzed in this chapter.

• 1. The Basic Game
  • 1.1 Card Counting
• 2. Playing against the House
  • 2.1 How Likely are the Distributions?
  • 2.2 Betting High and Low
  • 2.3 Reshuffling
• References
• Exercises and Projects

Let me begin by describing different variants of the game, depending on two parameters a and b:

Belonging to the sequential games with perfect information, this game can be displayed using extensive form, and be analyzed using backwards induction. Let us start with the extensive form, for which we will use a game tree instead of a game digraph. The situations where Ann has to move are uniquely determined by the cards Ann and Beth have so far, and the information whether one or both of them has stopped so far. This is encoded by writing the values of Ann's cards, then a hyphen, and then the values of Beth's cards. If Ann has stopped already, then the string of her cards is followed by a full stop sign, and the same for Beth. For example, "11.-12" means that Ann has two cards of value 1 each and has stopped already, and that Beth has one card of value 1 and one of vale 2. The same encoding is used for positions where Beth is supposed to move. We indicate whose turn it is to move by the color of the vertex.

However, we have as many games as we have different parameters a and b. Fortunately the structure of the game tree remains the same, only the probabilities for the different random moves depend on a and b (and are therefore omitted in the figure).

Note that some obviously bad move options are already omitted in the game tree. For instance, when a player has only one card and this card has the value 1 it doesn't make sense to stop (see the blue 1-1 or 1-2 positions). Also, if a player is behind, stopping doesn't make sense either (see the blue 11-12 position).

To analyze the game tree, we open the Excel sheet MiniBlackJack.xls at the sheet "Game". It contains the game tree, but since it is an Excel sheet the probabilities for the random moves are updated automatically as long as the user types the numbers a and b into the two grey fields. Even the backwards induction is done automatically and the blue and green cells under the leftmost vertex indicate the expected payoffs for Ann and Beth in the corresponding version MINI BLACKJACK(a,b) of the game when playing it. When trying it out, you will notice that the expected payoff for Ann is usually slightly negative.

I hope the reader agrees with me meanwhile that the process of backwards induction, though rather tedious and time-consuming, is in principle very easy. Easy enough for an Excel sheet to solve the game. But Excel can even more: It can compute all expected payoffs in the solutions for Ann and Beth in all possible combinations a and b of the parameters. The data is shown in the table at the bottom of the sheet "Game" and is copied here:

For instance, if you play MINI BLACKJACK(5,7) with five 1s and seven 2s, then the expected payoff for Ann at the very beginning is -0.02. The redder the cells, the higher the advantage for Beth. Note also that the ##### cells indicate that the game may not be played regularly until the decision, since there are too few cards in the stack.

Not only do the probabilities of the random moves, and therefore the expected payoffs change when the parameters a and b change, the strategies of the players might also change. Let me give an example. Look at the situation (a blue cell, labeled by "11-2." in the Excel sheet) where Ann has two cards of value 1 each, and Beth has only one card, a "2", and has already stopped. Now the question is whether Ann should take another card. Stopping means of course a draw with payoff of 0. If a=5 and b=2, then getting another card carries the expected payoff of 0.5, and is therefore favorable over stopping. If a=5 and b=3 the expected payoff for another card is still 0.2, but it becomes 0 for a=5 and a=4, and -0.14 for a=5 and b=5. To give another maybe less obvious example, look at the situation (a blue cell, labeled 2-2 on the Excel sheet) where both players have a "2" and Ann has to decide to get another card. If a=5 and b=5, stopping has an expected payoff of -0.25, but getting another card an expected payoff of -0.11. Therefore Ann would get another card in this situation. However, if a=5 and b=6, stopping has an expected payoff of -0.11, and getting another card an expected payoff of -0,17, therefore Ann would stop in this variant. Of course, in order to be able to play differently, the players must be aware that they are playing a different variant.

All these expectations for Ann are negative or zero. The second moving Beth has some advantage in this game.

In reality, we are not interested to play these 144 or more possible variants. The reason why they are discussed is that we need them to analyze the game MINI BLACKJACK without parameters. This game starts with a play of BLACKJACK(16,16). After the first play is decided, there are still a 1s and b 2s in the stack, and then a round/play of BLACKJACK(a,b) follows, and so on. Card counting means to keep track of all the cards drawn so far. A card counting player is always, and in particular at the beginning of each play, aware of the number of 1s and 2s still in the stack.

In real Black Jack, players are playing against the "House", represented by the dealer. The dealer is slightly handicapped insofar as he or she is not allowed to change his or her strategy depending on the distribution of the cards still in the stack. Instead the dealer decides whether to get a new card or not by fixed rules. In real Black Jack, this disadvantage of having to play unflexible is matched by the advantage for the dealer of having to move last in each round of card giving. This is not so much an advantage for the dealer, since he or she has to play mechanical anyway, but a disadvantage for the other players of not seeing the dealers final result.

To model this in MINI BLACKJACK, we declare that the dealer has to play always in the same way as he or she would optimally play in MINI BLACKJACK(16,16). To balance this disability to adapt, the dealer gets Beth's slightly better second player position.

Dealer Beth gets another card if she is behind ("11-1", "12-1", "2.-1", "12-1", "111-11", "12.-11", "12-2", "12-2"). Otherwise the dealer Beth gets a card if both have one card, a "2" each, and Ann has stopped ("2.-2"). The dealer stops in the remaining three cases of a draw ("11.-11", "2.-11", "11-2").

The "Dealer" sheet on the Excel sheet models this modified game. Of course, the dealer Beth doesn't have any options for her "moves" anymore, so the whole game could also be modeled as a 1-person game. Again different numbers a and b result in different expected payoffs for Ann, and again there is a table at the bottom of the page displaying Ann's payoffs (assuming best play for Ann) for all different distributions of 1s and 2s.

Red cells indicate an advantage for the dealer, Beth, and green cells an advantage for Ann. Some card distributions are very favorable for Ann, like those with many 1s (as a ≥ 13) and very few 2s (say b ≤ 2) with expected payoff larger than 0.5 for Ann. Other card distributions in the stack, like a=8 and b=8 with expected payoff of -0.07 for Ann, are slightly favorable for the dealer Beth. The more the distribution in the stack deviates from a equal distribution of the same number of 1s and 2s, the better for Ann. Probably this is due to the fact that Beth's fixed strategy is designed to work for the uniform distribution a=16 and b=16. And of course, in the case of very few 2s, it is always best to take another card as long as the sum is not yet 3, but the fixed dealer's strategy would not always do that.

Looking at Table 2, you may have the opinion that Ann has already an advantage, since there are more cells favorable to Ann, and the advantage for Ann, if there is one, seems to be in general larger than the advantage for Beth in the other cells. The average of all entries in Table 2 equals 0.116, so shouldn't Ann have an advantage? The answer is "No". And the reason is that the cells are not equally likely. It is rather likely to face a round of the game starting with eight 1s and eight 2s, but it is rather unlikely to start a round of the game with sixteen 1s and no 2 left. The more the card distribution deviates from equality between 1s and 2s, the more rare it is. And these rare situations are the ones where Ann has an advantage. This has to be taken into account.

In order to get a better idea of Ann's average chances in a round, we need to estimate how likely each of the subgames MINI BLACKJACK(a,b) is during a sequence of playing several rounds, starting with a stack of 16 1s and 16 2s. But instead of concentrating only on the distributions at the beginning of each new round, we look a little closer, look at all card distributions in the stack occurring in the different situations in the different rounds of the game. In other words, we look at the stack each time a card is dealt, not only at the beginning of the rounds.

The method used is a probability digraph. Let "a-b" denote the distribution of a 1s and b 2s. Since we start with "16-16", this situation occurs with probability 1. When removing one card, we get a "1" or a "2" with the same probability 16/32=1/2, therefore both distributions "15-16" and "16-15" have the same probability of 50%. But even when eliminating the next card is becomes obvious that some distributions are more likely than others. Since "16-15" would result in "16-14" and "15-15" with about equal probability (of 15/31 and 16/31), and since "15-16" would result in "15-15" and "14-16" with about equal probability, the distributions "14-16" and "16-14" occur roughly with probability 1/4 each, whereas "15-15" occurs with probability of about 1/2. The beginning of the probability digraph is displayed to the right. The probabilities of the arcs are green, and the probabilities of the situations in red. The simple but again tedious calculations of the rest are left to another Excel sheet. The numbers are given in Table 3. Note that the Start of the probability digraph is in the lower left corner of the table.

Remember that these are not the probabilities that a round is started with that many 1s and 2s if shuffling is only done after a round is finished. Since at every round 2 to 6 cards are dealed, no round would start with distribution 16-15, for instance. But for simplicity we will use these probabilities even in that case.

Our question is now: What is the expected payoff for Ann when one game MINI BLACKJACK(a,b) is played randomly, with the frequencies of Table 3. To go from the entries there to probabilities, we have to divide each entry by 27, so that all these probabilities add up to 1. Table 4 shows the products of these probabilities and the expected payoffs for Ann provided we start the round with that distribution from Table 2. The sum of all these values would be the expected payoff for Ann in the single random round. The sum equals -0.027. Ann loses about 3 cents per round when each round is played for $1. Not worse than Roulette, but still.

But Ann can do something about her fate. Except that Ann can count cards and react with her play to the situation, and the dealer Beth has to stick to the fixed given strategy, there is still another asymmetry between Ann and the dealer Beth which we have to take into account. In any round, Ann decides in advance how much to bet. Beth just has to match this.

This leads to a very simple winning strategy for Ann. Ann bets nothing unless a round starts with a distribution favorable for Ann, as shown in Table 2. Unfortunately this is not allowed in most casinos. If you sit at the table, you have to play. There is a minimum bet and a maximum bet. What Ann would do in that case is to bet low if Ann's expectation is negative, and high if it is positive. Therefore Ann must count cards for two reason: She always needs to now what distribution it is to adjust her playing strategy, as discussed above, but also to know what to bet.

To give an example explaining how this could be to the advantage of Ann, assume that four rounds are played with a stack. Let's also assume that the four different subgames have expected payoffs of -0.07, -0.03, 0.1, and -0.06. Although the positive expectation has higher absolute value than the three negative ones, the sum of the four values is still negative. But if Ann bets $1 in each of the negative rounds, but $2 in the positive round, her expected value of the whole sequence of games equals -0.07·1-0.03·1+0.1·2-0.06·1 = 0.04. By betting much higher in that round of positive expectations Ann would still increase her expected payoff. This urge to bet low in negative expectation rounds, and high in positive expectation rounds, is only limited by the Casino's rules about minimum and maximum bets, and in real life also by the player's desire not be revealed as card counter to the Casino, since the Casino might ban card counters from further play.

Table 5 shows the case of minimum bet of 1 and maximum bet of 20. The values are obtained from those of Table 4 by multiplication of the positive cells by 20. The sum of the values equals 0.197. Ann would on average win as much as 20 cents per round if there would not be reshuffling.

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